29th Annual American Mathematics Competitions AMC 8 TEACHERS’ MANUAL Instructions and Reporting Forms for School Contest Managers Tuesday, November 19, 2013 Please read this booklet thoroughly upon receipt. DATES TO REMEMBER AMC 10/AMC 12 - Tuesday, February 4, 2014 &/or Wednesday, February 19, 2014 AIME - Thursday, March 13, 2014 or Wednesday, March 26, 2014 USAMO & USAJMO - Tuesday & Wednesday, April 29-30, 2014 A M C 8 Dear AMC 8 Managers, Welcome to the 29th annual American Mathematics Competitions’ AMC 8. You and your students will find the 2013 AMC 8 an excellent learning experience. Please thoroughly read this manual for the rules and regulations for the AMC 8. Several days before the contest date fill out the School Identification Form and have the students complete the non-answer sections of the Answer Form (See Sections III and V) . We again have a “window” which will enable schools to participate officially in case of a school closing or an academic conflict with the official scheduled date. The alternate “window” dates are November 20-26. Further rules governing the use of the “window” are found on page 6 of this manual. The Edyth May Sliffe Award, recognizing distinguished mathematics teaching, will be continued in 2013. Further details for nominating a teacher from your school are on page 9. We will make every effort to process your results rapidly. Most schools will receive their results before the holiday break. On behalf of the Committee on the American Mathematics Competitions, I send my appreciation to you for all of your help with this contest. Sincerely, Dr. Steven R. Dunbar Director, MAA American Mathematics Competitions American Mathematics Competitions University of Nebraska - Lincoln | PO Box 880658 | 1740 Vine Street | Lincoln, NE 68588-0658 402.472.2257/phone | 402.472.6087/fax | [email protected] | maa.org/math-competitions Table of Contents Letter to Teachers................................................................................................................................................2 I. Points to Note......................................................................................................................................................3 II. Contents of The Package . .................................................................................................................................4 III. What Must Be Done Prior To November 19, 2013t���������������������������������������������������������������������������������������4 IV. Instructions For The Day of The Contest .......................................................................................................4 V. Instructions For Completing The School Identification Form.....................................................................5 VI. What To Do Following The Contest ...............................................................................................................5 VII. Report of Results.................................................................................................................................................5 VIII. Eligibility and Participation Schedule..............................................................................................................5 IX. Policy Statements................................................................................................................................................5 X. Intramural Awards..............................................................................................................................................6 Questionnaire to Help Us Serve You Better....................................................................................................7 XI. AMC 8 CERTIFICATION FORM 2013 .........................................................................................................8 XII. Sliffe Nomination Form ....................................................................................................................................9 XIII. Additional Bundles Form................................................................................................................................10 XIV. Proof of Intent to Pay.......................................................................................................................................10 XV. AMC 8 Rescoring Request Form....................................................................................................................11 XVI. AMC 8 Practice Questions..............................................................................................................................12 XVII. Publicity.............................................................................................................................................................22 XVIII. Letter to Parents................................................................................................................................................23 XIX. Certificate of Participation (facing page)......................................................................................................24 XX. Facsimile of the AMC 8 front cover...............................................................................................................25 AMC Sponsors and Contributors..................................................................................................................26 I. Points to Note Let Parents/Guardians know about this activity! NO CALCULATORS The Committee on the American Mathematics Competitions has decided that the use of any calculator will not be allowed. No problems on the contest will require the use of a calculator. The letter found on page 23 can be duplicated and sent home with students or handed out at parent/teacher conference time or open house. Block Scheduling or Other Conflicts Survey Questionnaire The Committee on the American Mathematics Competitions has decided that the contest may now be administered on more than one day during the window period. Please keep all student answer forms secured, and send everything back to us after all administrations have taken place. We are constantly trying to improve and streamline our processes. We would greatly appreciate it if you would take the time to answer the 8 questions found on page 7. Mark responses using the grid on the back side of the school ID Form. Thank you! Give your Student Scores a Boost! K-12 Teacher Membership You can find a sampling of good practice questions on pages 13-21 in this manual! Also, our Math Club Package provides more practice questions and resources to help teachers prepare students for the AMC 8. (Use the publications order form found on our website.) With registration, you automatically receive a K-12 Teacher Membership in the Mathematical Association of America with e-subscriptions to MAA magazines, discounts on books and meetings, and access to the Curriculum Inspirations resources. 3 II. Contents Of The Package IV. Instructions For The Day of The Contest Your contest package contains 1. Teachers’ Manual, 2. AMC 8 Report Envelope (marked with your school ID), 3. School Identification Form, 4. Contest bundles, 5. 10 Answer Forms for each bundle of 10 exams ordered, (Please check that the number of bundles corresponds to the number ordered. If it does not, call the AMC office as soon as possible.) 7. Contest Manager’s Envelope (contains a complimentary copy of the AMC 8 contest), 8. Complimentary materials, 9. Math Messenger. Do not open the contest bundle(s) until just before you are ready to hand the exams to the students. 1. Seat students so they are separated by an empty space, if possible. 2. Hand out the Answer Forms. 3. Provide a #2 lead pencil with a good eraser for each student. 4. Open the contest bundle(s), hand out the AMC 8 and inform the students that they are not to open the contest booklet until instructed to do so. 5. As you hand out the contest, tell the students to disregard the page which contains instructions for the contest manager. 6. Instruct students to read the entire front cover of the AMC 8. Allow 3 minutes to do this. 7. This 40 minute contest has twenty-five questions. When it is time to begin, tell students to unfold it to see all the problems and to be cautious about following the correct numeric sequence. 8. You will note that after the question there are five possible answers labeled A, B, C, D, and E. Only one answer is correct. Have your students mark their answers to these problems by the corresponding number on the answer form. 9. Tell your students to be AS CERTAIN AS POSSIBLE OF THEIR ANSWERS BEFORE THEY MARK THE FORM. Forms which contain imperfect erasures may not be read by the scanner that scores the contest. Tell your students to take special care to mark the answer in the row which corresponds to the problem number. 10.Remark to Teacher: Some students might wish to keep a personal record of their answers by circling their answers on the contest booklet before they are marked on the form. This procedure is acceptable provided it does not consume too much time. However, the official answers will be the ones blackened on the answer form. 11.Ask if there are any questions about procedures. Inform the participants that talking or asking questions during the contest is not allowed; they must do their own work. 12.Remind students that they have 40 minutes to complete the contest. Then tell them to BEGIN. 13.The contest should be proctored continuously as you would for any important contest. The proctor must warn any students whose eyes wander and disqualify any students caught copying answers or collaborating. Try to provide as quiet an environment as possible. 14.Announce when there are 15 minutes left and when there are 5 minutes left. 15.When time is up, tell the students to STOP and have them sign their name in the space provided on the Student Answer Form. Collect the Answer Forms and exams as quickly as possible. Inform the students that the contest and solutions may not be discussed with anyone outside of their contest administration either orally, via email, internet, copier or media of any type until after November 26. 16.Collect all the students’ Contest Booklets following the administraion of the contests. As the Answer Forms are handed in, check that each student’s name is on each one and bubbled in completely. 17.Please do not grade the answer forms. Send them to the AMC for grading. Tell the students that their scores will be available by late December. III. What Must Be Done Prior To November 19, 2013 1. Complete the following Pre-Examination processes associated with the AMC 8 Answer Forms: a. Have the students complete the following parts of the form. (The most common errors in the past have been filling in more than one circle in a column or not completely filling in the circles). (1) Last Name (2) First Name (3) Middle Initial (4) Section Letter (if applicable) For each math class participating in the AMC 8, you may assign a different section letter of the alphabet (A-Z). These letters are assigned at the Contest Manager’s discretion, and may be assigned by class, grade or ability group. Ten or more students should be assigned to each Section Letter. If you return results with Sections that have fewer than ten students, we will combine these Sections into a single Section. Certificates are given to the top 3 students in each section (see page 6-7). (5) Age/date of birth, grade and gender b. Collect the Answer Forms for redistribution on the contest date. c. Please make a final check to see if the forms have been marked correctly. 2. Review the rules found on the package(s) of the AMC 8. 3. Discuss with your students the questions and solutions from previous AMC 8’s so that they may see the type of questions to expect. 4. Review with your students the instructions on the cover of the 2013 AMC 8, found on page 25 of this manual. 5. The day before the contest, remind students of the time and place of the AMC 8. 6. Make arrangements to mail the AMC 8 REPORT ENVELOPE(s) by a trackable mail service. 7. Obtain a supply of: (a) #2 lead pencils with good erasers, and (b) unmarked scratch paper. Students may not use a calculator. Inform students that no aids are permitted other than scratch paper, graph paper, ruler, protractors, and erasers. No calculators are allowed. No problems on the contest will require the use of a calculator. 4 media of any type until after November 26. b. The contest booklets must be collected from the students. They may be returned to the students after November 26. V. Instructions For Completing The School Identification Form Several days before the contest date, please complete the following items on the School Identification Form. 1. Stamp or write your school name and complete school mailing address in the box in the upper-right hand corner. Please do not use address labels. 2. School ID -- Your School ID is on your AMC 8 Report Envelope. Copy your School ID in the empty boxes under the words “School ID.” Your school ID is 5 digits followed by a letter. 3. Mark the number of completed Answer Forms returned. Please do not count or return answer forms of students who were absent, blank forms or teacher “practice” forms. 4. Mark the proper circles for the date the contest was actually administered. 5. Be sure to complete the Survey on Page 7 by marking your answers on the back side of the ID form. VII. Report of Results The AMC 8 will be centrally scored at the AMC office at the University of Nebraska-Lincoln. Each school report includes the following data: 1. A list of the top 25 students in your school. 2. A report for each section with a minimum of 5 students. This report lists the score of each student. 3. One combined report for all sections with fewer than 5 students. 4. Tables of Item Difficulty for the participants in your school. 5. A 2014 AMC 10/AMC 12 Registration Form will be included with your results. The dates of the AMC 10/AMC 12 are Tuesday, February 4, 2014 and/or Wednesday, February 19, 2014. We recommend interested students study prior year AMC 10/AMC 12 exams. 6. The names of the top-scoring students and national award winners will be listed on the AMC website. They will be posted following the complete analysis of results. VI. What To Do Following The Contest 1. Place the School ID Form on top of the Answer Forms and insert in the Report Envelope. PLEASE NOTE: Notes: IF YOU DO NOT RECEIVE YOUR 2013 AMC 8 REPORT OF RESULTS BY JANUARY 15, 2014 PLEASE CALL THE AMC OFFICE at 800-527-3690. a. Please arrange the forms by section with each form facing the same direction with the same side up. REMOVE ALL PAPER CLIPS, RUBBER BANDS OR NOTE PAPER ATTACHED TO ANY ANSWER FORMS. PLEASE FAN THE FORMS SEPARATING ANY THAT MIGHT BE STUCK TOGETHER. b. Place no more than 200 forms in one envelope. c. Do not return blank forms or teacher “practice” forms. 2. Complete the Certification Form, page 8. If you did not nominate a teacher for the Sliffe award when you registered, please do so now on the Sliffe Award Nomination Form on page 9. 3. Ask your Principal or person with comparable title to sign the Certification Form. Place the Certification Form in the AMC 8 Report Envelope, seal and send it by a trackable mail service within 24 hours or as soon as possible after all administrations have taken place. ALL student answer forms returned for grading will be RECYCLED after February 15, 2014. VIII. Eligibility and Participation Schedule 1. Any student 14.5 years of age or younger on the day of the contest and not enrolled in grades 9, 10, 11, or 12 or equivalent is eligible to participate. 2. The contest may be given during regular mathematics classes, to a section of the students, or at one time to the entire group of students. Notes: Indicate the number of packages you are returning Notes: a. You should discuss academic integrity with your students if the contest is administered at several times or on more than one day, and all copies of the contest should be retained by the teacher until after all administrations have taken place. b. See Section IX for instructions relating to administering the contest on other dates. in the space provided on the Report Envelope. Affix adequate postage to the AMC 8 Report Envelope. The AMC 8 Student ANSWER FORMS sent to Lincoln for grading will not be returned. We charge $10 for rescoring a student’s individual contest. 4. Please note: After you have delivered all of the Answer Forms to the school office to be mailed, you may discuss the contest and solutions with your students. Remember that there will be schools taking the contest in other locations at different times, and some will be taking the contest on other “window” dates. a. Inform the students that the contest and solutions may not be discussed with anyone outside of their contest administration either orally, via email, internet, copier or IX. Policy Statements Statement 1 Official & Unofficial Administration of the AMC 8 Early Administration is NEVER permitted and will lead to a disqualification of the results. Such an administration would jeopardize the validity of all scores from other schools. Official Administration: If you are unable to give the contest on 5 November 19th because, for example, a.your school is closed, b.your school has an academic conflict, c.the class periods have been shortened due to an assembly or other reason, or d.the majority of your students will be on a field trip, not be eligible for state or national awards. A teacher or a school adminstrator may read the questions to the student and mark the answers as directed by the student. Please be sure to mark this information on the School Certification Form (page 8). Statement 6 then your school may use any day between November 20-26. The AMC 8 should be given officially on the first day that the school has available testing time, from Wednesday, November 20 through Tuesday, November 26. Please note: Because the contest may be given officially during the “window” days, as indicated above, you must collect the students’ contest booklets and instruct them not to discuss the contest questions and solutions outside their contest administration until after November 26. We define the term “contest administration” as a group of students who are taking the contest at the same time in the same room. Braille or Large Print Exams The AMC 8 is available in Braille and large print formats. Braille and large print exams must be ordered prior to October 29, 2013. Please contact us at [email protected] for more information. 1. The time limit set by the Committee on the American Mathematics Competitions for visually impaired students is 70 MINUTES. 2. Please note that a certified teacher or a school administrator may read the questions to the student and mark the answers as directed by the student. Unofficial administration after November 26 is permitted. Be sure to mark the exceptions box at the bottom of the certification form as indicated. The last date to take the contest unofficially is Friday, December 13th, 2013. Statement 7 English Language Learners English Language Learners may use a book or dual-language nontechnical dictionary between their native language and English. A student may use the dictionary only the first time that he/she takes the AMC 8. The dictionary must be given to the school contest manager to examine and retain for the 24-hour period preceding the test. The proctor must announce to other students that the student(s) has/have been given special permission to use the dictionary during the contest. Statement 2 Irregularities If it is clear to the Contest Manager or another proctor that a student has cheated, then the Manager must disqualify the student’s score and not submit that student’s AMC 8 ANSWER FORM for grading. If there is inconclusive evidence of cheating or an accusation without clear evidence, the Contest Manager must report this irregularity in writing when submitting the ANSWER FORMS for the school. This written report and the questionable ANSWER FORM(s) should be clearly identified, banded together and submitted along with the other ANSWER FORMS. Statement 8 Request for Student Names Policy The American Mathematics Competitions (AMC) receives requests from educational institutions and organizations for the names of high-scoring AMC 8 students. The AMC evaluates interested institutions and organizations individually for appropriateness. We do not collect addresses of students participating in the AMC 8, so we are not able to provide lists allowing direct contact of students. Instead, we provide legitimate educational institutions of all levels and professional and scholarly organizations with the postal address of school contest managers who have students meeting criteria such as high score or region of school. Then consistent with school policies and at the discretion of the school contest manager and the school administration, the contest manager can pass the organization’s information to students. Most organizations find this an effective and appropriate way to contact high-ability middle-school students. Statement 3 Follow-up Inquiries The results of this AMC 8 identify students with exceptional mathematical ability. To assure that the purpose is served, the Committee on the American Mathematics Competitions reserves the right to make inquiries about scores, to re-examine students and to disqualify scores if necessary. If the required security procedures are not followed, all scores from a school may be disqualified. Statement 4 Refund/Credit Policy If your school is unable to take the AMC 8, please use the materials as practice sets for the next year. WE CANNOT GIVE REFUNDS OR CREDITS AFTER THE CONTEST MATERIALS ARE SHIPPED. X. Intramural Awards School Certificate of Honor — awarded to schools with a team score (AMC 8, top 3 students scores) of 66 or greater. School Certificate of Merit — awarded to schools with a team score (AMC 8, top 3 students scores) between 50 and 65, inclusive. Certificate of Distinction — is given to all students who receive a perfect score. AMC 8 Winner Pin — given to the student(s) in each school with the highest score. Certificate for Outstanding Achievement — the top three Statement 5 Learning Disabilities The AMC allows schools to provide the same accommodations already established for timed mark-sense tests in a student’s written education program, including extra time up to a maximum of 70 minutes for the AMC 8. Students using additional time may 6 students in each of your assigned sections will receive, respectively, a gold, silver, or bronze certificate. In case of a tie, additional certificates will be included with your results. AMC 8 Distinguished Honor Roll Certificate — given to all students who score in (approximately) the top 1%. AMC 8 Honor Roll Certificate is given to all students who score in (approximately) the top 5%. AMC 8 Achievement Roll Certificate — given to all students in 6th grade and below who score 15 or above. Achievement Award Pins, Certificates of Merit, and neckties are available for purchase if your school wishes to recognize other outstanding students. Ordering information for these awards will be sent along with your school’s results. Questionnaire to Help Us Serve You Better Thank you in advance for taking the time to answer these questions for us. Answer to the best of your ability, and if you don’t know, just give us a “best estimate”. Please fill in your answers in the ovals provided on the back of the school ID form. Thank you! 1. Rate the difficulty of the AMC 8 problems 1-15 compared to the difficulty of your mathematics curriculum: 0)much easier 1) easier 2) about the same 3) harder 2. What level math classes take the AMC contest? Mark all that apply: 0) grade 8 1) grade 7 2) grade 6 5) only advanced students (all grades) 3) grade 5 & 4 4) much harder 4) Math Club (all ages) 3. What are the obstacles in your school to offering the AMC 8? Mark all that apply: 0) cost 1) paperwork/purchase order 2) scheduling 3) duration of contest 4) finding location 5) other 4. Would you be interested in receiving the AMC 8 Teachers’ Manual as a PDF or other electronic document? 0) very interested 1) interested 2) neutral 3) not interested 4) do not want 5. Would you be interested in a more condensed AMC 8 Teachers’ Manual? For example, only including instructions for the day of the contest and having other forms, like Additional Bundles, located online. 0) very interested 1) interested 2) neutral 3) not interested 4) do not want 6. Is the curriculum in your school or district adopting or moving toward the Common Core State Standards? 0)yes 1) no 2) not sure 7. Is the AMC 8 a good supplement to the Common Core State Standards in Mathematics’ topics? 0)yes 1) somewhat 2) no 3) not sure 8. Which of the following features would you be interested in having access to using your online AMC Contest Account? Mark all that apply: 0)ability to edit student information 1)prefilled student certificates 2)more payment options 3)ability to buy downloadable AMC Publications, like Math Club Packages, past contests, etc. 4)ability to build custom reports 5)I did not create an AMC Contest Account this year. 7 XI. AMC 8 CERTIFICATION FORM 2013 To be eligible, a student should be 14.5 years of age or younger on the day of the contest and not enrolled in grades 9, 10, 11, or 12 or equivalent. The AMC 8 must be administered by a teacher or an adult not associated with or related to any of the participants. The administration of the contest must take place in a public building (e.g. school, library, church). Please send all Answer Forms from your school or group at one time. For a school/home school, the Contest Manager and the Principal, Vice Principal, Headmaster, Home School Coordinator, or person with comparable title must sign this form and return it with your student Answer Forms. The Contest Manager must wait until all aspects of the contest process at the school have been completed. Certification by the principal, official or person with comparable title: a. I certify that the contest package(s) were retained in their sealed condition within an hour of the start of the administration of the contest. b. I accept for our school/home school the rules and procedures described on this page and pages 4-6, and accept that failure to follow these rules and procedures may result in DISQUALIFICATION from official standing of all scores from our school. Signature _________________________________________________ Time____________________________________ Title _________________________________________________ Date____________________________________ If your school did not administer the AMC 8 contest on November 19, 2013, please enter the date(s) given ___________________. If the contest was taken unofficially after the “window,” mark the exceptions box at the bottom and write the reason on the bottom of this page. Certification by the Contest Manager: I certify that the following statements are true or that, if there are any exceptions, I have checked the box below and have listed them below on the bottom of this page or on a separate sheet of paper. I understand that the absence of either signature from this form and a consideration of the exceptions may result in DISQUALIFICATION of all scores from our school. I understand that administering the AMC 8 before November 19, 2013 will lead to disqualification of our school’s results. 1. Only students in 8th grade or below, or ages 14.5 or below participated in the Contest. 2. The package of Contests was not opened until just before the administration of the Contest. 3. The participants were continuously monitored during the Contest, and they were separated by an empty space, if possible. 4. Participants had exactly 40 minutes working time. 5. No student was permitted to proctor the Contest. 6. I collected all of the students’ Contest booklets following the administration of the Contest. 7. After the Contest, the AMC 8 ANSWER FORMS were placed in the AMC 8 Report Envelope and kept secure. No changes were made in the answers. If the AMC 8 was given at more than one time, the Contest papers were kept secure until all administrations were over. 8. I have followed all the rules as stated in this Teachers’ Manual. DAY & DATE SIGNATURE_____________________________________ TEST WAS GIVEN ______________________ School Name _____________________________________ Telephone ______________________________ City ____________________________________________ State ___________ Email (please print clearly):__________________________ School Grade Level: _____ through _____ Zip ______________ PLEASE INDICATE THE EXACT NUMBER OF AMC 8 ANSWER FORMS RETURNED FOR GRADING: ___________ EXCEPTIONS If accommodations for a Learning Disability were made, please list student name(s): 8 XII. 2013 AMC 8 Sliffe Nomination Form for the Edyth May Sliffe Award for Distinguished Mathematics Teaching (please print all information clearly) The Sliffe Award is given each year to approximately fifty teachers from high-scoring AMC 8 schools. Winners of this award receive a recognition certificate, an award pin, a cash prize, and a one-year membership to the Mathematical Association of America. A special reception during the NCTM spring meeting honors the award winners. Nominate a teacher by filling out this form and submitting it to the AMC office or by filling out the online Sliffe nomination form found on the AMC website, maa.org/math-competitions (available under Teacher Resources, Sliffe Award). Name of School Nominated Mathematics Teacher:_ _______________________________________________ Only one name may be submitted from your 2013 AMC 8 registered school, and it should be the mathematics teacher who has been identified with the success of the students on the AMC 8. The teacher must be currently employed by the participating school from which the student answer forms have been submitted. It is perfectly acceptable to nominate yourself. However, past recipients of the Sliffe Award are ineligible. School ID #: ___ ___ ___ ___ ___ ___ Teacher’s Email:___________________________________________ School Name:_ ___________________________________________________________________________ City, State, & Zip:__________________________________________________________________________ School Telephone:_________________________________________________________________________ 9 XIII. Additional Bundles Form Please fill in the information below and FAX your order. The administrator, authorized person, or person ordering for the school agrees to pay the American Mathematics Competitions for the following materials: School Name _________________________________________________________ Phone # ______________ Address _____________________________________________________________ City______________________________________________ State_ ___________ Zip __________________ Name of person placing the order ______________________________________________________________ Email address ______________________________________________________________________________ AMC 8 Contest Bundles of ten ............................ #[email protected] $16/bundle =..............$ ___________ Solutions, Sets of ten (optional)................ #[email protected] $ 7/set =......................$ _ ____________ Postage/Handling Fee (see chart below) ............................................................................................... $___________ Total.............................................................................................................................................................. $___________ VISA/MC# : _______________________________________________ Exp. Date: _____________________ Sec. Code: ______ Name (Please Print):_________________________________________ Phone #_ _______________________________________ Email (needed for email receipt): ________________________________________________________________________________ AMC ORDERING -- TERMS 1. VISA & MasterCard accepted. 2. Make checks payable to: American Mathematics Competitions 3. PAYMENT IN U.S. FUNDS ONLY. 4. USA: $5.00 per bundle shipping charge* 5. OUTSIDE USA: $7.00 per bundle shipping charge* SCAN and EMAIL to: [email protected] OR FAX 402-472-6087 or send your order to: American Mathematics Competitions ATTN: AMC 8 Additional Bundles 1740 Vine Street Lincoln, NE 68508 *Orders for Additional Bundles will be accepted up to one week before the contest date. After that date, orders for Additional Bundles will be fulfilled by electronic delivery, and charged the Additional Copies rate of $3.00 per copy. XIV. Proof of Intent to Pay Billed P.O. Number ________________________________________________________________________ Complete billing address: __________________________________________________________________________ __________________________________________________________________________ 10 XV. AMC 8 Rescoring Request Form I would like to have the following student’s answer form re-scored. I understand that there is a $10.00 charge for each student answer form rescored. $ 10.00/each Student Name_____________________________________________________________ $_ ____________ Student Name_____________________________________________________________ $_ ____________ Student Name_____________________________________________________________ $_ ____________ Student Name_____________________________________________________________ $_ ____________ Grand Total _ _____________________________________________________________ $_ ____________ Teacher’s Name_____________________________________________School ID #__________________ School Name_____________________________________________Phone #_ __________________ Address ________________________________________________________________________ City______________________________________________________State_________Zip__________ Email where re-score should be sent _ ___________________________________________________ Method of Payment: (Rescores will not be processed until payment is received.) Check (US funds only) made payable and mailed with this form to the: AMERICAN MATHEMATICS COMPETITIONS 1740 Vine Street Lincoln, NE 68508 Charge to Visa/Mastercard#:___________________________________________________________ Name on card (print):_____________________________________________________________ Signature_______________________________________________________________________ Expiration Date: ________________ Sec. Code: _______ Telephone:_______________________ Email for electronic credit card receipt:_______________________________________________ SCAN and EMAIL completed form to: [email protected] or FAX to: 402-472-6087 11 XVI. AMC 8 Practice Questions Example Each of the following four large congruent squares is subdivided into combinations of congruent triangles or rectangles and is partially shaded. What percent of the total area is partially shaded? The original problem and choices from the 2011 AMC 8 contest (A) 12 1 2 (B) 20 (C) 25 (D) 33 1 3 (E) 37 1 2 2011 AMC 8, Problem #7— “Find the shaded portion of each square separately. ” Problem number Hint Solution Answer (C): The upper left and the lower right squares are each one-fourth shaded, for a total of one-half square. The shaded portions of the upper right and lower left squares make up one-half square. So the total shaded area is one full square, which is 25% of the total area. Difficulty: Medium SMP-CCSS: 2, 7 CCSS-M: 6G.1, 6RP.3C Standards for Math Practice Common Core State Standard 7 12 Solution from official solutions Difficulty, Percent correct Easy 100-80% Med Easy 80-60% Medium 60-40% Med Hard 40-20% Hard 20-0% AMC 8 Practice Questions Continued 12-01 Rachelle uses 3 pounds of meat to make 8 hamburgers for her family. How many pounds of meat does she need to make 24 hamburgers for a neighborhood picnic? (A) 6 (B) 6 2 3 (C) 7 1 2 (D) 8 (E) 9 2012 AMC 8, Problem #1— “Find how much meat the family would use.” Solution Answer (E): Rachelle needs 24 8 = 3 times the amount of meat for the picnic than she would use for her family. So she needs 3 × 3 = 9 pounds of meat. OR Set up a proportion to compare the two ratios of pounds of meat to number of hamburgers. x 3 = 8 24 Solving for x, 8x = 72, so x = 9 pounds of meat. Difficulty: SMP-CCSS: 1. Make sense of problems and persevere in solving them. CCSS-M: 6.RP.3. Use ratio and rate reasoning to solve real-world and mathematical problems. 1 13 AMC 8 Practice Questions Continued 12-04 Peter’s family ordered a 12-slice pizza for dinner. Peter ate one slice and shared another slice equally with his brother Paul. What fraction of the pizza did Peter eat? (A) 1 24 (B) 1 12 (C) 1 8 (D) 1 6 (E) 1 4 2012 AMC 8, Problem #4— “Find the fraction of the total pizza for a whole slice.” Solution 1 Answer (C): The whole slice that Peter ate was 12 of the pizza. His half of the second slice was half of of the pizza. The fraction of the pizza that Peter ate was 1 1 2 1 3 1 + = + = = . 12 24 24 24 24 8 Difficulty: SMP-CCSS: 1. Make sense of problems and persevere in solving them. CCSS-M: 5.NF.1. Add and subtract fractions with unlike denominators. 4 14 1 12 , or 1 24 , AMC 8 Practice Questions Continued 12-06 A rectangular photograph is placed in a frame that forms a border two inches wide on all sides of the photograph. The photograph measures 8 inches high and 10 inches wide. What is the area of the border, in square inches? (A) 36 (B) 40 (C) 64 (D) 72 (E) 88 2012 AMC 8, Problem #6— “What is the width and height of the frame?” Solution Answer (E): The width of the frame is 10 + 2 + 2 = 14 inches, and its height is 8 + 2 + 2 = 12 inches. It encloses an area of 14 × 12 = 168 square inches. The photograph occupies 10 × 8 = 80 square inches of that area, so the area of the border itself is 168 − 80 = 88 square inches. Difficulty: SMP-CCSS: 1. Make sense of problems and persevere in solving them. CCSS-M: 3.MD.7. Relate area to the operations of multiplication and addition. 6 15 AMC 8 Practice Questions Continued 12-07 Isabella must take four 100-point tests in her math class. Her goal is to achieve an average grade of at least 95 on the tests. Her first two test scores were 97 and 91. After seeing her score on the third test, she realized that she could still reach her goal. What is the lowest possible score she could have made on the third test? (A) 90 (B) 92 (C) 95 (D) 96 (E) 97 2012 AMC 8, Problem #7— “What should Isabella’s total score for all 4 tests be with an average of 95?” Solution Answer (B): To achieve an average grade of 95 on the four tests, Isabella must score a total of 4 × 95 = 380 points. She scored a total of 97 + 91 = 188 points on her first two tests, so she must score a total of at least 380 − 188 = 192 points on her last two tests. Because she can score at most 100 on her fourth test, she must have scored at least 92 on her third test. OR Isabella’s first score was 95 + 2, her second score was 95 − 4, and her fourth score can be at most 95 + 5. Because she can still achieve an average score of 95, her third score must have been at least 95 − 2 + 4 − 5 = 92. Difficulty: SMP-CCSS: 1. Make sense of problems and persevere in solving them. CCSS-M: 6.SP.3. Recognize that a measure of center for a numerical data set summarizes all of its values with a single number, while a measure of variation describes how its values vary with a single number. 7 16 AMC 8 Practice Questions Continued 12-08 A shop advertises that everything is “half price in today’s sale.” In addition, a coupon gives a 20% discount on sale prices. Using the coupon, the price today represents what percentage discount off the original price? (A) 10 (B) 33 (C) 40 (D) 60 (E) 70 2012 AMC 8, Problem #8— “What is the price of an item after both discounts are applied?” Solution Answer (D): The price of an item costing $d after both discounts are applied is .80(.50d) = .40d, a discount of 60% off the original price. Difficulty: SMP-CCSS: 1. Make sense of problems and persevere in solving them. CCSS-M: 7.RP.3. Use proportional relationships to solve multistep ratio and percent problems. 8 17 AMC 8 Practice Questions Continued 12-09 The Fort Worth Zoo has a number of two-legged birds and a number of four-legged mammals. On one visit to the zoo, Margie counted 200 heads and 522 legs. How many of the animals that Margie counted were two-legged birds? (A) 61 (B) 122 (C) 139 (D) 150 (E) 161 2012 AMC 8, Problem #9— “How many birds were four-legged?” Solution Answer (C): All 200 heads belonged to animals with at least two legs, accounting for 400 of the 522 legs. The additional 122 legs belonged to four-legged mammals, each of which had two additional legs. So Margie saw 122 2 = 61 four-legged mammals and 200 − 61 = 139 birds. Difficulty: SMP-CCSS: 1. Make sense of problems and persevere in solving them. CCSS-M: 4.OA.3. Solve multistep word problems posed with whole numbers and having whole-number answers using the four operations, including problems in which remainders must be interpreted. 9 18 AMC 8 Practice Questions Continued 12-17 A square with an integer side length is cut into 10 squares, all of which have integer side length and at least 8 of which have area 1. What is the smallest possible value of the length of the side of the original square? (A) 3 (B) 4 (C) 5 (D) 6 (E) 7 2012 AMC 8, Problem #17— “What is the least possible value for the area of the original square?” Solution Answer (B): The area of the original square is a square number that is more than 8, so 16 is the least possible value for the area of the original square. Its side has length 4. Two possible ways of cutting the square are shown below: Difficulty: SMP-CCSS: 1. Make sense of problems and persevere in solving them. CCSS-M: 7.G.2. Draw (freehand, with ruler and protractor, and with technology) geometric shapes with given conditions. 17 19 AMC 8 Practice Questions Continued 12-20 What is the correct ordering of the three numbers 9 23 , in increasing order? (A) (C) (E) 9 23 9 23 7 21 7 5 < 21 < 19 5 7 < 19 < 21 5 9 < 19 < 23 (B) (D) 5 19 5 19 5 7 19 , 21 , < < 7 21 9 23 and < < 9 23 7 21 2012 AMC 8, Problem #20— “Use a common denominator for all three fractions.” Solution Answer (B): Using a common denominator, 7 9 189 7 9 Also 21 = 161 483 and 23 = 483 , so 21 < 23 . 5 19 = 105 399 and 7 21 = 133 399 , so 5 19 < 7 21 . OR Comparing each fraction, 5 7 9 19 < 21 < 23 . 7 21 = 1 3, 5 19 < 5 15 = 1 3, and 9 23 > 9 27 = 1 3, so the correct increasing order is Difficulty: SMP-CCSS: 1. Make sense of problems and persevere in solving them. CCSS-M: 6.NS.7. Understand ordering and absolute value of rational numbers. 20 20 AMC 8 Practice Questions Continued 12-24 A circle of radius 2 is cut into four congruent arcs. The four arcs are joined to form the star figure shown. What is the ratio of the area of the star figure to the area of the original circle? 4−π (A) π 1 (B) π (C) √ 2 π (D) π−1 π (E) 3 π 2012 AMC 8, Problem #24— “Translate the star into the circle and create a square concentric to the circle.” Solution Answer (A): Translate the star into the circle so that the points of the star coincide with the points on the circle. Construct four segments connecting the consecutive points of the circle and the star, creating a square concentric to the circle. The area of the circle is π(2)2 = 4π. The square is made up of four congruent right triangles with area so the area of the square is 4 × 2 = 8. The area inside the circle but outside the square is 4π − 8. 1 2 (2×2) = 2, This is also the area inside the square but outside the star. So, the area of the star is 8 − (4π − 8) = 16 − 4π. The ratio of the area of the star figure to the area of the original circle is 16−4π = 4−π 4π π . Difficulty: SMP-CCSS: 1. Make sense of problems and persevere in solving them. CCSS-M: 7.G.4. Know the formulas for the area and circumference of a circle and use them to solve problems. 24 21 XVII. Publicity If possible the sample Contest Announcement and Results Announcement news releases should be prepared and distributed to the newspapers, radio and television stations in your region. See the AMC website for statistics and figures from the the 2013 contest available in December. (School or School District) FOR IMMEDIATE RELEASE (School) STUDENTS INVITED TO WORLD-WIDE COMPETITION ( # ) students at (School) participated in the 29th annual American Mathematics Competitions 8, held on Tuesday, November 19, 2013. They competed for local and national student and school awards. The contest, which covers middle school mathematics, is given at participating schools. Its purpose is to spur interest in mathematics and develop talent through the excitement of friendly competition at problem solving in a timed format. In 2012 over 150,000 students from over 2,300 schools participated in the AMC 8 contest including ( # ) students from ( # ) schools in (State). The AMC 8, first offered in 1985, is an annual project sponsored by The Mathematical Association of America, and is located at the University of Nebraska – Lincoln. The AMC receives support from the Akamai Foundation, Academy of Applied Sciences, American Mathematical Association of Two-Year Colleges, American Mathematical Society, American Statistical Association, Art of Problem Solving, Awesome Math, Casualty Actuarial Society, Conference Board of the Mathematical Sciences, the D.E. Shaw Group, Jane Street Capital, Math for America, Mu Alpha Theta, Andy and Laurie Okun, Pi Mu Epsilon, the Society for Industiral and Applied Mathematics and W.H. Freeman. The Contest is given across the USA, Canada, and in many schools abroad. Details concerning the 2013-2014 AMC contests for Middle School and High School are available on the AMC web site: maa.org/math-competitions. Entries for the AMC 10/12 2014 contest close on January 24, 2014 for the February 4th exam, and on February 7, 2014 for the February 19nd contest. For further information contact the AMC -- email us at [email protected] 22 Fall 2013 Dear Parent or Guardian: On November 19, 2013 your son or daughter will participate in the 29th annual American Mathematics Competitions 8 (AMC 8) contest being offered in your school. Last year the AMC 8 contest involved over 150,000 students worldwide. With these contests, there are awards in each school for the student with the highest score, certificates for high-scoring students in each school, and state-wide awards. The AMC 8 contest can lead to other more selective math contests, even all the way to the United States of America team sent to the International Mathematical Olympiad, the premier international high school level mathematical problem solving contest. But the real rewards come from challenging each student with mathematics that is new, different, and “outside of the box.” The problems on the contest are difficult, but designed to be within the reach of students. Even so, if your son or daughter managed to solve only one or two problems, this should be an encouragement to learn more, because these problems are meant to be more challenging than students routinely encounter in their mathematics courses. Mathematics is increasingly important in our technological and scientific age. Taking enough mathematics in school is the gateway to jobs and careers of all kinds, even those that are not explicitly mathematical, scientific or technological. We hope that by offering these contests, we can challenge and inspire students to want to learn more mathematics. We hope that your son or daughter enjoys the contests and will continue to take mathematics courses in middle school, high school and beyond. Sincerely, Dr. Steven R. Dunbar Director, MAA American Mathematics Competitions American Mathematics Competitions University of Nebraska - Lincoln | PO Box 880658 | 1740 Vine Street | Lincoln, NE 68588-0658 402.472.2257/phone | 402.472.6087/fax | [email protected] | maa.org/math-competitions 23 American Mathematics Competitions CERTIFICATE Awarded to for participating in the American Mathematics Contest 8 (AMC 8) __________________________________ 2013 Steven R. Dunbar, Director American Mathematics Competitions __________________________________ Margie Raub Hunt, Chair AMC 8 Subcommittee American Mathematics Competitions CERTIFICATE Awarded to for participating in the American Mathematics Contest 8 (AMC 8) __________________________________ Steven R. Dunbar, Director American Mathematics Competitions 2013 __________________________________ Margie Raub Hunt, Chair AMC 8 Subcommittee XIX. Certificate of Participation (facing page) (for reproduction, page 24) XX. Facsimile of the AMC 8 front cover American Mathematics Competitions 29th Annual AMC 8 American Mathematics Contest 8 Tuesday, November 19, 2013 INSTRUCTIONS 1. DO NOT OPEN THIS BOOKLET UNTIL YOUR PROCTOR TELLS YOU. 2. This is a twenty-five question multiple choice test. For each question, only one answer choice is correct. 3. Mark your answer to each problem on the AMC 8 Answer Form with a #2 pencil. Check the blackened circles for accuracy and erase errors and stray marks completely. Only answers properly marked on the answer form will be graded. 4. There is no penalty for guessing. Your score is the number of correct answers. 5. Only scratch paper, graph paper, rulers, protractors, and erasers are allowed as aids. Calculators are NOT allowed. No problems on the test require the use of a calculator. 6. Figures are not necessarily drawn to scale. 7. Before beginning the test, your proctor will ask you to record your information on the answer form. 8. You will have 40 minutes to complete the test once your proctor tells you to begin. 9. When you finish the exam, sign your name in the space provided on the answer form. The Committee on the American Mathematics Competitions reserves the right to re-examine students before deciding whether to grant official status to their scores. The Committee also reserves the right to disqualify all scores from a school if it determines that the required security procedures were not followed. The publication, reproduction or communication of the problems or solutions of the AMC 8 during the period when students are eligible to participate seriously jeopardizes the integrity of the results. Dissemination via copier, telephone, email, internet or media of any type during this period is a violation of the competition rules. After the contest period, permission to make copies of individual problems in paper or electronic form including posting on web pages for educational use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear the copyright notice. © 2013 Mathematical Association of America 25 The American Mathematics Competitions are Sponsored by The Mathematical Association of America – MAA .................................................... www.maa.org The Akamai Foundation ........................................................................................www.akamai.com Contributors Academy of Applied Sciences – AAS...................................................................................................www.aas-world.org American Mathematical Association of Two-Year Colleges – AMATYC................................................. www.amatyc.org American Mathematical Society – AMS...................................................................................................... www.ams.org American Statistical Association – ASA................................................................................................ www.amstat.org Art of Problem Solving – AoPS........................................................................................www.artofproblemsolving.com Awesome Math ........................................................................................................................... www.awesomemath.org Casualty Actuarial Society – CAS........................................................................................................... www.casact.org Conference Board of the Mathematical Sciences – CBMS................................................................. www.cbmsweb.org The D.E. Shaw Group..................................................................................................................... www.deshaw.com Jane Street Capital..................................................................................................................www.janestreet.com Math For America............................................................................................................. www.mathforamerica.org Mu Alpha Theta – MAT................................................................................................................. www.mualphatheta.org Pi Mu Epsilon – PME.......................................................................................................................... www.pme-math.org Society for Industrial and Applied Math (SIAM)..................................................................................www.siam.org W.H. Freeman........................................................................................................................... www.whfreeman.com

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